Incursive and hyperincursive systems, fractal machine and anticipatory logic

Abstract

This paper shows why incursive and hyperincursive systems are computing anticipatory systems. Examples are reviewed for fractal cellular automata, finite difference equation systems and Boolean Logic. Weak and strong anticipation deal with prediction and built anticipation, respectively. Internal and external time may be at the basis of some computing anticipatory systems. In view of computing differential equation systems, discrete equation systems depend on new variables given by intervals of time Δt and/or space (Δx,Δy,Δz), differential equation systems only depending on the current space-time. Due to the discrete and recursive representation of continuous systems, time and space shifts appear, which must be compensated by anticipatory effects. Some well-known algorithms are explicitly anticipatory as the predictor-corrector algorithm and the finite difference implicit method. These anticipatory algorithms are related to computing incursive systems.